Square Root of 311

√311 at a glance

Exact value

√311

Decimal (10 places)

17.6351920885

Rounded

17.6 · 17.64 · 17.635

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.635192

Prime factorization

311

Cube root

6.775169

How to simplify √311

311 is a prime number, so its only factors are 1 and 311. There is no perfect-square factor to pull out, which means √311 is already in its simplest radical form.

The square root of any prime is irrational. If √311 were a fraction a/b in lowest terms, then a² = 311b², so 311 would divide a — and then 311 would divide b too, contradicting “lowest terms.” That is why the decimal 17.6351920885 is only a rounded value.

Where √311 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √311 lies between 17 and 18. 311 is 22 above 289 and 13 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.6286 (0.04% low)
• Tangent from 17, i.e. 17 + 22 ÷ 34: 17.6471 (0.07% high)
• Tangent from 18, i.e. 18 − 13 ÷ 36: 17.6389 (0.02% high)

For √311 the tangent at 18 wins, missing by only 0.0037. Tangent estimates shine when the number sits close to a perfect square — here 311 is just 13 below 324.

Finding √311 with the Babylonian method

If a guess is too big, 311 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√311) in one step.

Start from the nearest whole number, 18 (18² = 324):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √311 = 17.6351920885 to every decimal shown.

√311 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √311 the pattern is [17; 1, 1, 1, 2, 1, 6, 3, 17, 3, 6, 1, 2, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √311 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 311y² = 1. Its smallest solution in positive whole numbers is x = 16,883,880, y = 957,397.

√311 in geometry and everyday measurements

• A square patio or deck of 311 square feet is about 17.64 ft (17 ft 8 in) on each side, so edging all the way around takes 4 × √311 ≈ 70.5 ft.
• 311 is not a sum of two whole-number squares — 311 is itself a prime that is one less than a multiple of 4, which rules that out — so √311 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √311 as its space diagonal.

Square roots near √311 and related roots

• The cube root of 311 is about 6.775169.
• Squaring undoes the root: (√311)² = 311, while 311² = 96,721 — the number whose square root is 311.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.